Binomial sums related to rational approximations to $ζ(4)$

Details Binomial sums related to rational approximations to $ζ(4)$ Binomial sums related to rational approximations to $ζ(4)$

2003-11-12

arxiv.org/abs/math/0311196v1

Mat. Zametki 75:4 (2004), 637--640 (Russian); English transl., Math. Notes 75:4 (2004), 594--597

Mathematics

Classical Analysis and ODEs

5 pages, AmSTeX

Scientific paper

10.1023/B:MATN.0000023341.93824.

For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,...$, are denominators of rational approximations to $\zeta(4)$ (see math.NT/0201024). We use Andrews's generalization of Whipple's transformation of a terminating ${}_7F_6(1)$-series and the method from math.NT/0311114.

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